1. Introduction
Firstly, fuzzy vector subspace was introduced by Katsaras and Liu [1] . Then its pro- perties and characters were investigated (see [2] [3] [4] [5] , etc). The dimension of a fuzzy vector space was defined as a n-tuple by Lowen [6] . Subsequently, it was defined as a non-negative real number or infinity by Lubczonok [5] , and proved that the for- mula
(1)
is valid under certain conditions, where
and
are fuzzy vector spaces. Recently, basis and dimension of a fuzzy vector space were redefined as a fuzzy set and a fuzzy natural number by Shi and Huang [7] , respectively. Under the definitions, more pro- perties of (crisp) vector spaces were correct in fuzzy vector spaces.
In this paper, we generalize the results in [7] to L lattice, and prove that some for- mulas still hold in the lattice L. In particular, we present the definition of L-fuzzy vector subspace and its -fuzzy dimension. The L-fuzzy dimension of a finite dimensional fuzzy vector subspace is a fuzzy natural number. We prove that (1) holds without any re- stricted conditions and
holds.
2. Preliminaries
Given a set
and a completely distributive lattice L, we denote the power set of
and the set of all L-fuzzy sets on
(or L-sets for short) by
and
, respec- tively . For any
, we denote the cardinality of
by
.
An element
in L is called a prime element if
implies
or
.
in L is called co-prime if
implies
or
[8] . The set of non- unit prime elements in L is denoted by
. The set of non-zero co-prime elements in L is denoted by
.
The binary relation
in L is defined as follows: for
,
if and only if for every subset
, the relation
always implies the existence of
with
[9] .
is called the greatest minimal family of
in the sense of [10] , denoted by
, and
. Moreover, for
, we define
and
. In a completely distri- butive lattice
, there exist
and
for each
, and
(see [10] ).
In [10] , Wang thought that
and
. In fact, it should be that
and
.
Throughout this paper,
denotes a completely distributive lattice, and
is a crisp vector space. We often do not distinguish a crisp subset
of
and its cha- racteristic function
.
If
and
, we can define
![]()
![]()
Some properties of these cut sets can be found in [11] - [16] .
In [17] Shi introduced the concept of L-fuzzy natural numbers(denoted by
), defined their operations and discussed the relation of
-cut sets. We simply recall as follows: for any
,
,
(1) ![]()
(2) ![]()
(3) For any
and
, it follows that ![]()
3. L-Fuzzy Vector Subspaces
Definition 3.1. L-fuzzy vector subspace is a pair
where
is a vector space on field
,
is a map with the property that for any
, we have
.
In this definition, when
, L-fuzzy vector subspace is exactly the fuzzy vector subspace defined in [1] . We denote the family of L-fuzzy vector subspaces by
.
Let
be a member of
, we denote
![]()
.
We can obtain some properties of
analogous to fuzzy vector subspaces as follows.
Theorem 3.2. Let
be a member of
, then
(1) ![]()
(2) For any ![]()
The prove is trivial and omitted.
Remark: Since
is a completely distributive lattice, the property that if
, then
not holds for
. This can be seen from the following example.
Example 3.3. Let
be a completely distributive lattice with four elements as fol- lows.
![]()
Let
be an L-fuzzy vector subspace on
where
is defined by
![]()
We can easily check
is an L-fuzzy vector subspace on
. Suppose that
and
, then
This example illustrates for L-fuzzy vector subspace
, ![]()
Theorem 3.4. Let
be a vector space,
and
. Then the follow- ing statements are equivalent:
(1)
is an L-fuzzy vector subspace.
(2) (a) ![]()
(b) ![]()
(3) For any
and
, where
is a finite natural number, we have
![]()
The prove is trivial and omitted.
In the following paper, the vector spaces we discuss are finite-dimensional. For their L-fuzzy vector subspaces, the following observation will be useful.
Remark: Let
be a member of
. Suppose that
. Since
is finite-dimensional vector space, denotes
, then
is a finite subset of L.
In the fact, let
be a basis of
, then
. Suppose that
is infinite, then for all
, the total number of
is infinite. Since
is a basis of
, we have
. Again since
is finite, the total number of
is also finite. It contradicts with the hypothesis. Therefore
is a finite subset of
with at most
values;
values which can be attained at the vectors of
and the maximum which is attained at 0.
Theorem 3.5. Let
be a vector space,
and
. Then the follow- ing statements equivalent:
(1)
is an L-fuzzy vector subspace.
(2) For all
,
is a vector space.
(3) For all
,
is a vector space.
(4) For all
,
is a vector space.
(5) For all
,
is a vector space.
(6) For all
,
is a vector space.
Proof. We prove
and
, the others can be proved analogously.
We show that
is a vector space as follows. Suppose that
, then
and
, i.e.
.
Since
be an L-fuzzy vector subspace, then
, we have
, this means
. Therefore
is a vector space.
Suppose that for all
,
is a vector space. Let
and
. Since
is a vector space, then
if and only if
. We have
![]()
Therefore
is an L-fuzzy vector subspace.
Suppose that
, then
and
. Since
, then
. Because
is an L-fuzzy vector subspace, we can have
, this implies
. Thus
is a vector space.
Let
and
. Since
is a vector space, then
if and only if
. We have the following implications.
![]()
Therefore
is an L-fuzzy vector subspace.
Theorem 3.6. Let
be a vector space,
be a map,
, and for all
. Then the following statements equivalent:
(1)
is an L-fuzzy vector subspace.
(2) For all
,
is a vector space.
Proof.
Suppose that
, then
, i.e.
. Since for all
and
is an L-fuzzy vector subspace, we can know
, this implies
. Therefore
is a vector space.
Suppose that for all
,
is a vector space. Let
and
. Since
is a vector space, then
if and only if
. We have
![]()
Therefore
is an L-fuzzy vector subspace.
We can define the operations between two L-fuzzy vector subspaces analogous to fuzzy vector subspaces.
Definition 3.7. Let
be two L-fuzzy vector subspaces on
. Define the intersection of
and
to be
. Define the sum of
and
to be
where
is defined by for all ![]()
![]()
Definition 3.8. Let
be two members of
and
. We define the direct sum of
and
to be
where
is defined by for all ![]()
![]()
Theorem 3.9. Let
be two members of
on
. We have
(1)
is a member of
on
.
(2)
is a member of
on
.
The proof of the theorem is trivial and it is omitted.
Theorem 3.10. Let
and
be the members of
. We have
(1) For all
, ![]()
(2) For all
, ![]()
(3) For any
, ![]()
(4) For any
, ![]()
Proof. The proofs of (1) and (2) are easy by the definition of
and the pro- perties of L-fuzzy sets.
(3) For any
, we have
![]()
(4) By the definition of the sum of L-fuzzy vector subspaces, for any
we have
![]()
Theorem 3.11. Let
and
be two members of
. Suppose that for any
, we have
. Then
(1) ![]()
(2) ![]()
The prove is trivial and omitted.
4. Fuzzy Dimension of L-Fuzzy Vector Subspaces
Definition 4.1. Let
be the family of L-fuzzy natural number. The map
is defined by
![]()
is called the L-fuzzy dimensional function of the L-fuzzy vector subspace
, and
is called the L-fuzzy dimension of
, it is an L-fuzzy natural number. We usually use another form of
as follows.
![]()
Theorem 4.2. For each
and
, we have
![]()
Proof. For any
, let
. Obviously
. Next we show that
Suppose that
and
, then there
exists
such that
. In this case,
which implies
. Thus we have
![]()
This completes the proof.
Theorem 4.3. Let the pair
be a member of
. Then for any ![]()
![]()
If
for all
, then
![]()
In particular,
for any
.
Proof. In order to prove
. Suppose that
, then
. Since
is a preserve-union map, there is
and
Because
, thus
. There- fore
.
is obvious. Moreover, we can obtain that
from the definition of ![]()
In order to prove for any
, we only need to show
. Since the set
is finite, for any
we have
![]()
Therefore ![]()
Theorem 4.4. Let
be a member of
. Then
![]()
In particular,
for any ![]()
Proof.
can be proved from the following implications.
![]()
Let
. In order to show
, we need to show that
Suppose that
. Since the number of
is finite, then when
the number of
is finite, denotes
, where
for any
Thus
Since
, then we have
Further we have
. Thus for any
![]()
Therefore for any
, ![]()
is obvious. We show that
in the follow- ing implications.
![]()
Theorem 4.5. Let
and
be two L-fuzzy vector subspaces. Then the following equality holds
![]()
Proof. We denote the sum of
by
. From Theorem 11, we know that
is a L-fuzzy vector subspace. By the properties of L-fuzzy na- tural numbers, Theorem 12 and the dimensional formulation of vector spaces, we know for any
,
![]()
Therefore ![]()
Definition 4.6. Suppose that
is an L-fuzzy vector subspace. A map
is called an L-fuzzy linear transformation, if it satisfies the following conditions:
(1)
is a linear map on
.
(2) For all
, ![]()
Theorem 4.7. Suppose that
is an L-fuzzy vector subspace,
is an L-fuzzy linear transformation on
, then
and
are L-fuzzy vector subspaces.
The prove is trivial and omitted.
Theorem 4.8. Suppose that
is an L-fuzzy vector subspace,
is an L-fuzzy linear transformation, then
![]()
Proof. Suppose that
is a linear transformation on (crisp) vector spaces
, then the equality
holds. Hence, for all
we have
![]()
Since
is a linear transformation on
, we have
![]()
Therefore
.
5. Conclusion
In this paper, L-fuzzy vector subspace is defined and showed that its dimension is an L-fuzzy natural number. Based on the definitions, some good properties of crisp vector spaces are hold in a finite-dimensional L-fuzzy vector subspace. In particular, the
equality
holds without any restricted conditions. At the same time,
holds.
Acknowledgements
The authors would like to thank the reviewers for their valuable comments and sug- gestions.
Fund
The project is by the Science & Technology Program of Beijing Municipal Commission of Education (KM201611417007), the NNSF of China (11371002), the academic youth backbone project of Heilongjiang Education Department (1251G3036), the foundation of Heilongjiang Province (A201209).